4 problems

Khintchine's moving-target conjecture. If $psi:\mathbb{N}\tomathbb{R}_{geq 0}$ is decreasing and $sum_{q=1}^{infty}psi(q)=infty$, then

Let $\mathbf{x}=(x_n)_{n=1}^{\infty}$ be a sequence in $[0,1]$, let $\psi:\mathbb{N}\to[0,\infty)$, and define … Here $\|\cdot\|$ denotes distance to the nearest integer, and $\mu$…

Let $c8\colon \mathbb{N}\to[0,1/2]$, let $\gamma\in[0,1]$, and let $\mu$ be the natural measure on the middle third Cantor set. For…

Let $T_2:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ be the doubling map, let $\mu$ denote Lebesgue measure on $\mathbb{R}/\mathbb{Z}$, let $\psi:\mathbb{N}\to[0,\infty)$, and l…